Q6: Degree of $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = \left(\frac{d^2y}{dx^2}\right)^2$

The degree is defined only when the equation is a polynomial in its derivatives: no derivative under a root, in a denominator, or inside a function such as $\sin$ or $\log$⁠.

Here both sides are whole-number powers of brackets of derivatives, so this holds and (d) is ruled out.

The degree is the highest power of the highest order derivative. The order here is 2, from $\frac{d^2y}{dx^2}$⁠.

Watch out: the power 3 belongs to a bracket of $\frac{dy}{dx}$⁠, a first order derivative. Expanding it gives $\left(\frac{dy}{dx}\right)^6$⁠, but that does not decide the degree.

Watch out: taking the square root of both sides gives

That form is not a polynomial in the derivatives, so it cannot be used for the degree. The wrong option $\frac{3}{2}$ comes from it.